A tuner accuracy is ±0.5 cent. Does that mean that it is highly sensitive or just decently sensitive for pitch accuracy?
According to Wiki., a one cent change is imperceptible to most humans, so half of that is going to be a pretty accurate parameter change. One semitone (say from E to F, or A to Bb) is easily recognised, but when that interval of a minor second is split into 100 different 'notes', most wouldn't spot note 77 from 78 as being different/out of tune. Halve that difference, and I'd say it's certainly accurate enough for me! Just how much more accurate would one need it to be?
I do my own tuning using TuneLab software. I try to get all strings within 0.5 cent of perfect. Then I check for beats, especially very slow ones, like 2-3 seconds. It sounds good when I play, so I must be doing something right. At the first sign of beating I touch up the tuning. (It helps to have a Dampp-Chaser installed.)
FWIW, I've been playing - and restoring - a 1943 Baldwin Acrosonic and have just upgraded to a 1959 Baldwin model M grand. The grand is in amazing condition for its age but for some reason was a few Hertz below A440. I'm about in the middle of tuning it, which is what prompted my search for what kind of tolerance is acceptable.
It depends on for what purpose. For tuning a violin or guitar, you should be fine. For tuning a rigid-pitch instrument with composite notes (like an accordion or organ) tuning two notes independently with 0.5 cent error each that are supposed to be sounding in unison (either with the same frequency or a fixed ratio like 2:1 or even the 3:2 of an organ's quint registers) is not going to sound well.